NC Algorithms for Popular Matchings in One-Sided Preference Systems and Related Problems

TL;DR

This paper introduces the first NC algorithms for popular matchings without ties, maximum-cardinality popular matchings, and a variant of the stable matching problem, advancing parallel algorithms in preference-based matching problems.

Contribution

It provides the first NC algorithms for popular matchings without ties, maximum-cardinality popular matchings, and a new approach for finding the next stable matching.

Findings

First NC algorithms for popular matchings without ties

NC algorithm for maximum-cardinality popular matchings

NC algorithm for next stable matching problem

Abstract

The popular matching problem is of matching a set of applicants to a set of posts, where each applicant has a preference list, ranking a non-empty subset of posts in the order of preference, possibly with ties. A matching M is popular if there is no other matching M' such that more applicants prefer M' to M. We give the first NC algorithm to solve the popular matching problem without ties. We also give an NC algorithm that solves the maximum-cardinality popular matching problem. No NC or RNC algorithms were known for the matching problem in preference systems prior to this work. Moreover, we give an NC algorithm for a weaker version of the stable matching problem, that is, the problem of finding the "next" stable matching given a stable matching.